1 Outline Part I: Theory steady state enzyme kinetics: a new - PDF document

A New 'Microscopic' Look at Steady-state Enzyme Kinetics Petr Kuzmi č BioKin Ltd. http://www.biokin.com SEMINAR: University of Massachusetts Medical School Worcester, MA April 6, 2015 Steady-State Enzyme Kinetics 1 1

Outline Part I: Theory steady state enzyme kinetics: a new approach Part II: Experiment inosine-5’-monophosphate dehydrogenase Steady-State Enzyme Kinetics 2 2

Enzyme kinetic modeling and its importance WHAT CAN ENZYME KINETICS DO FOR US? macroscopic microscopic laboratory molecular mathematical measurement mechanisms model EXAMPLE : Michaelis-Menten (1913) MECHANISM: initial rate maximum! substrate and enzyme form a reactive complex , which decomposes into products rectangular and regenerates the enzyme hyperbola catalyst [substrate] v = V m S/(S+K m ) Steady-State Enzyme Kinetics 3 The main purpose of enzyme kinetics is to elucidate microscopic molecular mechanisms of enzyme action and inhibition. It is a remarkable feat that we can go from observing events on a macroscopic scale (i.e. plate-reader well) to making inference about events that evolve on a molecular scale . Another use of enzyme kinetics, no less important, is to measure quantitatively the strength enzyme binding with various ligands (substrates, inhibitors ), as well as the reaction rates. 3

Two types of enzyme kinetic experiments UV/Vis absorbance, A 1. “reaction progress” method: slope = “rate” time 2. “initial rate” method: rate = dA/d t @ t = 0 [Subtrate] Steady-State Enzyme Kinetics 4 1. “reaction progress” method: • mix enzyme and reactants • monitor some experimental signal over time • build mathematical models of the reaction time course • see which of the models fits best 2. “initial rate” method: • mix enzyme and reactants • monitor some experimental signal over time • compute the slope (reaction rate) at time = 0 • repeat at various concentrations of reactants • build mathematical models of the initial rates changing with initial concentrations • see which of the models fits best 4

The steady-state approximation in enzyme kinetics Two different mathematical formalisms for initial rate enzyme kinetics: k chem << k d.S 1. “rapid equilibrium” approximation k chem << k d.P k chem ≈ k d.S 2. “steady state” approximation k chem > k d.S or k chem ≈ k d.P k chem > k d.P Steady-State Enzyme Kinetics 5 Analysis of initial rates in enzyme kinetics usually proceeds while invoking one of two theoretical formalisms, or approximations: • “rapid equilibrium” approximation • slow chemistry • fast ligand dissociation 2. “steady state” approximation • fast or slow chemistry • it is a more general approach 5

Importance of steady-state treatment: Therapeutic inhibitors MANY ENZYMES THAT ARE TARGETS FOR DRUG DESIGN DISPLAY “FAST CHEMISTRY” Example : Inosine-5’-monophosphate dehydrogenase from Cryptosporidium parvum chemical step: fast hydride transfer T. Riera et al. (2008) Biochemistry 47 , 8689–8696 Steady-State Enzyme Kinetics 6 Arguably the “holy grail” of enzyme kinetics, in the context of therapeutic inhibition, is to understand the microscopic rate constants for enzyme-inhibitor interactions. However, such detailed understanding is not possible for “fast enzymes”, unless we actually invoke the steady-state approximation. The reason is that specifically the binding of inhibitors to enzyme-substrate complexes or enzyme-product complexes cannot be properly quantified without having a firm grasp of all microscopic rate constants in the given mechanism. (This is not true for “competitive” interactions, where the inhibitor binds to the free enzyme . For those interactions, it is enough to know the actual “on” and “off” rate constants.) Here is an example of an enzyme that has “fast” chemistry and so we must study the inhibition kinetics under the steady-state approximation. 6

Steady-state initial rate equations: The conventional approach The King-Altman method conventionally proceeds in two separate steps: Step One: Derive a rate equation in terms of microscopic rate constants Step Two: Rearrange the original equation in terms of secondary “kinetic constants” EXPERIMENT: • Measure “kinetic constants” (K m , V max , ...) experimentally. • Compute micro-constant (k on , k off , ...) from “kinetic constants”, when possible. Steady-State Enzyme Kinetics 7 7

Steady-state initial rate equations: Example 1. postulate a particular kinetic mechanism: 2. derivation (“Step One”): 3. rearrangement (“Step Two”): micro-constants “kinetic” constants Details: Segel, I. (1975) Enzyme Kinetics , Chapter 9, pp. 509-529. Steady-State Enzyme Kinetics 8 •Whether or not the steady-state approximation is necessary depends only on the relative magnitude of the chemical rate constants (red) vs. the dissociation rate constants (blue). •Remember: The microscopic rates of the association steps (black) depend on concentrations and therefore have no upper bound ! •After the first step of the derivation, the rate equation consists of microscopic rate constants. Those are usually not accessible to direct measurement in initial-rate experiments. •In the second step, the micro-constants are grouped in various ways into “kinetic” constants such as K m nd k cat . •Very soon we will see that this second step of the derivation is not always possible. In contrast, the first step is always possible. 8

Several problems with the conventional approach 1. Fundamental problem: Step 2 (deriving “K m ” etc.) is in principle impossible for branched mechanisms. 2. Technical problem: Even when Step 2 is possible in principle, it is tedious and error prone . 3. Resource problem: Measuring “kinetic constants” (K m , K i , ...) consumes a lot of time and materials . Steady-State Enzyme Kinetics 9 1. Fundamental problem: Step 2 (rearrangement) is in principle impossible for branched mechanisms. The means: There can be no K m , V max , etc. derived for branched mechanisms. We can get a rate equation in terms of micro-constants, but then we are stuck. 2. Technical problem: Even when Step 2 is possible in principle, it is tedious and error prone. These algebraic derivations typically are for “hard-core” kineticists only. It would be nice to automate all derivations and relegate this task to a machine. 3. Economics or resource management problem: Measuring “kinetic constants” (K m , k cat , ...) consumes a lot of time and materials. Plus, it is not very clear how to convert “kinetic constants” to micro-constants. It would be nice to do a global fit , to extract as many micro-constants as possible. 9

A solution to the fundamental problem TURN THE CONVENTIONAL APPROACH ON ITS HEAD: CONVENTIONAL APPROACH: • Measure “kinetic constants” (K m , V max , ...) experimentally, when they do exist. • Compute micro-constant (k on , k off , ...) from “kinetic constants”, when possible. THE NEW APPROACH: • Measure micro-constant (k on , k off , ...) experimentally. • Compute “kinetic constants” (K m , V max , ...), when they do exist. Steady-State Enzyme Kinetics 10 Advantages of the new approach: 1. Rate equations formulated in terms of micro-constants always exist . Rate equations in terms of “kinetic constants” do not always exist (branched mechanisms). 2. In some cases the best-fit values of micro-constants are actually their “true” values . Caveat : in other cases the best-fit micro-constants are only “apparent” values. But Even if we cannot estimate the “true” values (model redundancy, see below), we can often at least estimate either the lower or the upper bound for a given micro- constant. Disadvantages of the new approach: 1. In most cases the model formulated in terms of micro-constants is overparametrized (redundant) . However, model redundancy can be dealt with in numerous ways: - make educated guesses based on literature reports; - supply estimates from independent rapid-kinetic measurements; - construct a minimal (“reduced”) kinetic model, with fewer steps 10

A solution to the technical / logistical problem USE A SUITABLE COMPUTER PROGRAM TO AUTOMATE ALL ALGEBRAIC DERIVATIONS INPUT : OUTPUT : Kuzmic, P. (2009) Meth. Enzymol. 467 , 247-280. Steady-State Enzyme Kinetics 11 • The DynaFit software is available free of charge (to all academic researchers and students) from the BioKin website. • It has been cited in close to 900 journal articles up to this point, most of them in the journal Biochemistry , followed by JBC . • The “King-Altman” method in DynaFit has been beefed up in the last month or so, specifically to facilitate the IMPDH collaboration with Liz Hedstrom’s group at Brandeis. 11